1.02a Indices: laws of indices for rational exponents

230 questions

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OCR MEI C1 2013 June Q7
5 marks Moderate -0.8
  1. Express \(125\sqrt{5}\) in the form \(5^t\). [2]
  2. Simplify \(10 + 7\sqrt{5} + \frac{38}{1 - 2\sqrt{5}}\), giving your answer in the form \(a + b\sqrt{5}\). [3]
Edexcel C1 Q1
4 marks Easy -1.8
  1. Express \(\frac{21}{\sqrt{7}}\) in the form \(k\sqrt{7}\). [2]
  2. Express \(8^{-1}\) as an exact fraction in its simplest form. [2]
Edexcel C1 Q7
9 marks Moderate -0.3
$$\text{f}(x) = \frac{(x-4)^2}{2x^{\frac{1}{2}}}, \quad x > 0.$$
  1. Find the values of the constants \(A\), \(B\) and \(C\) such that $$\text{f}(x) = Ax^{\frac{3}{2}} + Bx^{\frac{1}{2}} + Cx^{-\frac{1}{2}}.$$ [3]
  2. Show that $$\text{f}'(x) = \frac{(3x+4)(x-4)}{4x^{\frac{3}{2}}}.$$ [6]
Edexcel C1 Q3
4 marks Easy -1.3
  1. Solve the equation $$x^{\frac{3}{2}} = 27.$$ [2]
  2. Express \((2\frac{1}{4})^{-\frac{3}{2}}\) as an exact fraction in its simplest form. [2]
Edexcel C1 Q1
3 marks Moderate -0.8
Find the value of \(y\) such that $$4^{y + 3} = 8.$$ [3]
Edexcel C1 Q1
3 marks Easy -1.8
Evaluate \(49^{\frac{1}{2}} + 8^{\frac{1}{3}}\). [3]
OCR C1 Q1
4 marks Easy -1.8
  1. Express \(\frac{21}{\sqrt{7}}\) in the form \(k\sqrt{7}\). [2]
  2. Express \(8^{-1}\) as an exact fraction in its simplest form. [2]
OCR C1 Q3
4 marks Easy -1.3
  1. Solve the equation $$x^{\frac{3}{2}} = 27.$$ [2]
  2. Express \((2\frac{1}{4})^{-\frac{3}{2}}\) as an exact fraction in its simplest form. [2]
OCR C1 Q1
4 marks Moderate -0.5
Find the value of \(y\) such that $$4^{y+1} = 8^{2y-1}.$$ [4]
OCR C1 Q1
3 marks Easy -1.8
Evaluate \(49^{\frac{1}{2}} + 8^{\frac{2}{3}}\). [3]
OCR MEI C1 Q1
2 marks Easy -1.8
Make \(r\) the subject of the formula \(A = \pi r^2(x+y)\), where \(r > 0\). [2]
OCR MEI C1 Q4
4 marks Moderate -0.5
Make \(a\) the subject of \(3(a + 4) = ac + 5f\). [4]
OCR MEI C1 Q7
3 marks Easy -1.2
Simplify \(\frac{(4x^5 y)^3}{(2xy^2) \times (8x^{10}y^4)}\). [3]
OCR MEI C1 Q9
4 marks Moderate -0.8
Rearrange the equation \(5c + 9t = a(2c + t)\) to make \(c\) the subject. [4]
OCR MEI C1 Q1
4 marks Easy -1.3
Evaluate the following.
  1. \(200^0\) [1]
  2. \(\left(\frac{25}{9}\right)^{-\frac{1}{2}}\) [3]
OCR MEI C1 Q2
5 marks Easy -1.3
  1. Evaluate \(\left(\frac{1}{27}\right)^{\frac{2}{3}}\). [2]
  2. Simplify \(\frac{(4a^2c)^3}{32a^4c^7}\). [3]
OCR MEI C1 Q4
5 marks Easy -1.3
  1. Evaluate \((0.2)^{-2}\). [2]
  2. Simplify \((16a^{12})^{\frac{3}{4}}\). [3]
OCR MEI C1 Q5
4 marks Easy -1.8
Find the value of each of the following.
  1. \(\left(\frac{5}{3}\right)^{-2}\) [2]
  2. \(81^{\frac{1}{4}}\) [2]
OCR MEI C1 Q6
4 marks Easy -1.8
  1. Evaluate \(\left(\frac{1}{5}\right)^{-2}\). [2]
  2. Evaluate \(\left(\frac{8}{27}\right)^{\frac{2}{3}}\). [2]
OCR MEI C1 Q8
5 marks Easy -1.3
  1. Evaluate \(9^{-\frac{1}{2}}\). [2]
  2. Simplify \(\frac{(4x^4)^3 y^2}{2x^4 y^5}\). [3]
OCR MEI C1 Q10
5 marks Easy -1.3
  1. Evaluate \(\left(\frac{9}{16}\right)^{-\frac{1}{2}}\). [2]
  2. Simplify \(\frac{(2ac^2)^3 \times 9a^2c}{36a^4c^{12}}\). [3]
OCR MEI C1 Q11
4 marks Easy -1.8
  1. Write down the value of each of the following.
    1. \(4^{-2}\) [1]
    2. \(9^0\) [1]
  2. Find the value of \(\left(\frac{64}{125}\right)^{\frac{4}{3}}\). [2]
OCR MEI C1 Q1
5 marks Easy -1.2
  1. Express \(\frac{81}{\sqrt{3}}\) in the form \(3^k\). [2]
  2. Express \(\frac{5 + \sqrt{3}}{5 - \sqrt{3}}\) in the form \(\frac{a + b\sqrt{3}}{c}\), where \(a\), \(b\) and \(c\) are integers. [3]
OCR MEI C1 Q2
5 marks Easy -1.8
  1. Simplify \((5a^2b)^2 \times 2b^4\). [2]
  2. Evaluate \(\left(\frac{4}{16}\right)^{-1}\). [1]
  3. Evaluate \((16)^{\frac{3}{2}}\). [2]
OCR MEI C1 Q4
5 marks Easy -1.2
  1. Find the value of \(144^{-\frac{1}{2}}\). [2]
  2. Simplify \(\frac{1}{5 + \sqrt{7}} + \frac{4}{5 - \sqrt{7}}\). Give your answer in the form \(\frac{a + b\sqrt{7}}{c}\). [3]